I am trying different methods for numerical integration of an oscillatory integral of the following form:

$$\Psi(x) = \int_0^\infty \frac{A(s)\sin(sx)}{s(A(s)^2+c^2s^2)}ds$$

Where $$A(s) = 1 + 2(1- \cos(s))~, \quad c\in[0,10]$$

I am comparing the error resulting from trapezoidal integration of a Fourier interpolation and a composite Simpson's rule for the function but need help in dealing with the bounds of the integral.

I tried changing the variable of integration to $t = \frac{1}{s+1}+1$ but it caused the integral to oscillate immensely for values of x as low as $x=10^{-4}$

If I were to truncate the upper bound, how would I go about choosing a large enough number and determining the error from doing so?

  • $\begingroup$ Do a change of variables, something like $s=\tan t$. The limits for $t$ will be from $0$ to $\pi/2$ $\endgroup$
    – Andrei
    Commented Nov 23, 2021 at 20:31


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